A NEW PROOF OF THE SINGULAR CONTINUITY OF THE MINKOWSKI $?$ -FUNCTION

Let n be a natural number and M a set of n x n-matrices over the nonnegative integers such that M generates a finite multiplicative monoid. We show that if the zero matrix 0 is a product of matrices in M, then there are M_1, ..., M_{n^5} in M with M_1 *s M_{n^5} = 0. This result has applications in automata theory and the theory of codes. Specifically, if X subset Sigma^* is a finite incomplete code, then there exists a word w in Sigma^* of length polynomial in sum_{x in X} |x| such that w is not a factor of any word in X^*. This proves a weak version of Restivo’s conjecture.

A matrix is binary if each of its entries is either 0 or 1. The binary rank of a nonnegative integer matrix A is the smallest integer b such that A = BC, where B and C are binary matrices, and B has b columns. In this paper, bounds for the binary rank are given, and nonnegative integer matrices that attain the lower bound are characterized. Moreover, binary ranks of nonnegative integer matrices with low ranks are determined, and binary ranks of nonnegative integer Jacobi matrices are estimated.

We prove an asymptotic estimate for the number of m × n non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer-feasible flows in a network. Similarly, we estimate the volume of the polytope of m × n non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems, and hence can be computed efficiently. As a corollary, we show that if row sums R = (r1, …, rm) and column sums C = (c1, …, cn) with r1 + ⋯ + rm = c1 + ⋯ + cn = Nare sufficiently far from constant vectors, then, asymptotically, in the uniform probability space of the m × nnon-negative integer matrices with the total sum N of entries, the event consisting of the matrices with row sums R and the event consisting of the matrices with column sums C are positively correlated.

In this paper it is shown that an eventuallynonnegative matrix A whose index of zero is less than or equal to one, exhibits manyof the same combinatorial properties as a nonnegative matrix. In particular, there is a positive integer g such that A g is nonnegative, A and A g have the same irreducible classes, and the transitive closure of the reduced graph of A is the same as the transitive closure of the reduced graph of Ag. In this instance, manyof the combinatorial properties of nonnegative matrices carryover to this subclass of the eventuallynonnegative matrices. 1. Introduction. The Perron-Frobenius Theorem for irreducible nonnegative matrices has spawned a wealth of interesting ideas in the study of nonnegative ma- trices. Graph-theoretic spectral theory of matrices continues to develop, and in this paper we are interested in extending many of these ideas to the class of eventually nonnegative matrices. The relationship between the combinatorial structure of a non- negative matrix and its spectrum, eigenvectors, and Jordan structure is surprisingly elegant and beautiful, as well as useful. Surveys of results of this type can be found in Berman and Plemmons (1), Hershkowitz (5), and Schneider (13). Friedland (3), Handelman (4), Zaslavsky and Tam (16), and Zaslavsky and Mc- Donald (15) have looked at extending some of these combinatorial ideas to eventually positive matrices and eventually nonnegative matrices. In their study they found examples of eventually nonnegative matrices for which the relationship between the combinatorial structure of the matrix, and its spectrum, eigenvectors, and Jordan structure was inconsistent with that of nonnegative matrices. For example, there are irreducible eventually nonnegative matrices for which the spectral radius is a multiple eigenvalue. Even when the spectral radius is a simple eigenvalue of an irreducible even- tually nonnegative matrix, the associated eigenvector need not be positive. Moving to properties of reducible eventually nonnegative matrices, we see that the combinatorial spectral properties exhibited by reducible nonnegative matrices need not carry over to eventually nonnegative matrices. In this paper we show that it is really the contributions from the nilpotent part of an eventually nonnegative matrix that determine whether or not the combinatorial structure will be an accurate predictor of its spectral properties. In Section 3, we show that if A is an eventually nonnegative matrix that is nonsingular, or if all the Jordan blocks in the Jordan form of A associated with the eigenvalue zero are 1 × 1, then there is a positive integer g such that the transitive closure of the reduced graph

Strong shift equivalence and shear adjacency of nonnegative square integer matrices

Algorithmic aspects of arithmetical structures

The one-to-one correspondence between the set of plane partitions withr rows andm columns and the set of matrices of nonnegative integers with the same numbers of rows and columns has been constructed.

Hölder type matrix inequalities of Pate, Blakley, and Roy extended to the inner product of Frobenius

Let S be a countable topological space. If S is indexed by the positive integers, then real functions on S may be considered as sequences and are subject to infinite matrix transformation. After Henriksen and Isbell in [3], a point s in S is called a heavy point if there exists a regular matrix which sums every bounded real-valued function on S continuous at s. It is clear that the notion of a heavy point is independent of the order chosen for S. Some properties of heavy points are given in [3 ]. For example, sequential limit points are heavy points and isolated points (in fact, points whose complements are C*embedded) are not heavy points. In addition Henriksen and Isbell observe that not all heavy points are sequential limit points. The latter phenomenon was the motivation for the present paper. Corresponding to each positive regular matrix A we define a topology on the positive integers by taking neighborhoods of 1 to be sets whose characteristic functions are A-summable to 1. Denote the resulting space by NA. The point 1 is a heavy point in NA. Some basic properties of these spaces are examined. Also, we determine the class of matrices A for which 1 is not a sequential limit point in NA. Let A = (ank) be an infinite real matrix. The A-transform of the real sequence x = { xk } is the sequence { Sk ankXk } if these sums exist. Let c and m denote the Banach spaces of convergent sequences and bounded sequences, respectively, with jjx|I =supk| X . Let CA = {x: AxEc}. The matrix A is said to sum the members of CA or any subset of CA. Define the functional limA on CA by limAx=lim Ax. The matrix A is called conservative if CCCA and regular if in addition lim =limA on c. It is well known that A is regular if and only if limn ank=O for all k, limn Ekank=l1 and SUpn Zk lank! < o. The matrix A is called positive if ank _ 0 for all n, k. For a set T of positive integers let x(T) denote the characteristic function of T; (X(T))k =1 for k E T, 0 otherwise. For a topological space S let C*(S) denote the Banach space of bounded real-valued continuous functions on S with ||x|| =sup { I x(s)|: sES}. As observed in [3], if S is countable and completely regular, then S is 0-dimensional (i.e. the ring r of open and

We present a formalism to investigate directionality principles in evolution theory for populations, the dynamics of which can be described by a positive matrix cocycle (product of random positive matrices). For the latter, we establish a random version of the Perron-Frobenius theory which extends all known results and enables us to characterize the equilibrium state of a corresponding abstract symbolic dynamical system by an extremal principle. We develop a thermodynamic formalism for random dynamical systems, and in this framework prove that the top Lyapunov exponent is an analytic function of the generator of the cocycle. On this basis a fluctuation theory for products of positive random matrices can be developed which leads to an inequality in dynamical entropy that can be interpreted as a directionality principle for the mutation and selection process in evolutionary dynamics.

The asymptotic number of non-negative integer matrices with given row and column sums

We count mxn non-negative integer matrices (contingency tables) with prescribed row and column sums (margins). For a wide class of smooth margins we establish a computationally efficient asymptotic formula approximating the number of matrices within a relative error which approaches 0 as m and n grow.

We study the calculation of the volume of the polytope Bn of n × n doubly stochastic matrices (real nonnegative matrices with row and column sums equal to one). We describe two methods. The first involves a decompos ition of the polytope into simplices. The second involves the enumeration of “magic squares”, that is, n × n nonnegative integer matrices whose rows and columns all sum to the same integer. We have used the first method to confirm the previously known values through n = 7. This method can also be used to compute the volumes of faces of Bn For example, we have observed that the volume of a particular face of Bn appears to be a product of Catalan numbers. We have used the second method to find the volume for n = 8, which we believe was not previously known.

Let $C\geq 2$ be a positive integer. Consider the set of $n\times n$ non-negative integer matrices whose row sums and column sums are all equal to $Cn$ and let $X=(X_{ij})_{1\leq i,j\leq n}$ be uniformly distributed on this set. This $X$ is called the random contingency table with uniform margin. In this paper, we study various asymptotic properties of $X=(X_{ij})_{1\leq i,j\leq n}$ as $n\to\infty$.

An upper bound for the permanent of a nonnegative matrix